Lect24_AngMomentII - The Angular Momentum Operator The...

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Lecture 24: Angular Momentum II PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore The Angular Momentum Operator The angular momentum operator is defined as: It is a vector operator: According to the definition of the cross-product, the components are given by: P R L r r r ! = z z y y x x e L e L e L L r r r r + + = y z x ZP YP L ! = z x y XP ZP L ! = x y z YP XP L ! =
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Commutation Relations The commutation relations are given by: These are not definitions, they are just a consequence of [ X , P ]= i h These commutation relations are very important Any three operators which obey these relations are considered as ‘generalized angular momentum operators’ Compact notation: z y x L i L L h = ] , [ x z y L i L L h = ] , [ y x z L i L L h = ] , [ l l h L i L L jk k j ! = ] , [ 0 if any two indices are the same 1 cyclic permutations of x,y,z (or 1,2,3) -1 cyclic permutations of z,y,x (or 3,2,1) l jk ‘Levi Cevita tensor’ Summation over l is implied Simultaneous Eigenstates You will see that: Where: This means that simultaneous eigenstates of L 2 and L z exist Let: We want to find a and b . 0 ] , [ 2 = x L L 0 ] , [ 2 = y L L 0 ] , [ 2 = z L L 2 2 2 2 z y x L L L L + + = b a a b a L , , 2 = b a b b a L z , , =
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Algebraic solution to angular momentum eigenvalue problem
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Lect24_AngMomentII - The Angular Momentum Operator The...

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